Stochastic partial differential equations on the sphere. Andriy - PowerPoint PPT Presentation

Stochastic partial differential equations on the sphere. Andriy Olenko La Trobe University, Australia Monash Workshop on Numerical Differential Equations and Applications February 13, 2020 The talk is based on joint results with P. Broadbridge, D. Omari (La Trobe), V. Anh (QUT, Swinburne), N. Leonenko (Cardiff, UK), A.D. Kolesnik (Institute of Mathematics, Moldova), Y.G Wang (UNSW) A.Olenko SPDEs on sphere 1 / 32

TALK OUTLINE Introduction to CMB data 1 Spherical random fields 2 A.Olenko SPDEs on sphere 2 / 32

TALK OUTLINE Introduction to CMB data 1 Spherical random fields 2 Evolution of CMB: SPDEs 3 A.Olenko SPDEs on sphere 2 / 32

TALK OUTLINE Introduction to CMB data 1 Spherical random fields 2 Evolution of CMB: SPDEs 3 Numerical studies 4 A.Olenko SPDEs on sphere 2 / 32

TALK OUTLINE Introduction to CMB data 1 Spherical random fields 2 Evolution of CMB: SPDEs 3 Numerical studies 4 A.Olenko SPDEs on sphere 2 / 32

Introduction to CMB data Image credit: NASA / WMAP Science Team A.Olenko SPDEs on sphere 3 / 32

CMB is the remnant heat left over from the Big Bang; Predicted by Ralph Alpher and Robert Herman in 1948; Observed by Arno Penzias and Robert Wilson in 1965; Hundreds of cosmic microwave background experiments have been conducted to measure CMB; Most detailed space mission to date was conducted by the European Space Agency, via the Planck Surveyor satellite (in the range of frequencies from 30 to 857 GHz). A.Olenko SPDEs on sphere 4 / 32

Missions Image credit: https://jgudmunds.wordpress.com A.Olenko SPDEs on sphere 5 / 32

Next Generation Missions Next Generation Explorer: CMB-S4 (Sponsored by Simons Foundations, NSF and US Department of Energy) A.Olenko SPDEs on sphere 6 / 32

What does CMB data look like? 13.77 billion year old temperature fluctuations that correspond to the seeds that grew to become the galaxies Current CMB data are at 5 arcminutes resolution on the sphere. Contains 50,331,648 data collected by Planck mission. A.Olenko SPDEs on sphere 7 / 32

Research directions Direction 1: Stochastic modelling and developing new spherical inference tools: high frequency asymptotics, Minkowski functionals, R´ enyi functions Direction 2: Evolution of CMB: SPDEs Direction 3: Practical statistical analysis of CMB data: R package rcosmo A.Olenko SPDEs on sphere 8 / 32

Research directions Direction 1: Stochastic modelling and developing new spherical inference tools: high frequency asymptotics, Minkowski functionals, R´ enyi functions Direction 2: Evolution of CMB: SPDEs Direction 3: Practical statistical analysis of CMB data: R package rcosmo A.Olenko SPDEs on sphere 9 / 32

Spherical random fields The standard statistical model for CMB is an isotropic random field on the sphere S 2 T = { T ( θ, ϕ ) = T ω ( θ, ϕ ) : 0 ≤ θ < π, 0 ≤ ϕ < 2 π, ω ∈ Ω } . CMB can be viewed as a single realization of this random field. We consider a real-valued second-order spherical random field T that is continuous in the mean-square sense. The field T can be expanded in the mean-square sense as a Laplace series l ∞ � � T ( θ, ϕ ) = a lm Y lm ( θ, ϕ ) , l =0 m = − l where { Y lm ( θ, ϕ ) } represents the complex spherical harmonics. A.Olenko SPDEs on sphere 10 / 32

The random coefficients a lm in the Laplace series can be obtained through inversion arguments in the form of mean-square stochastic integrals � π � 2 π T ( θ, ϕ ) Y ∗ a lm = lm ( θ, ϕ ) sin θ d θ d ϕ. 0 0 The field is isotropic if − l ′ ≤ m ′ ≤ l ′ . l ′ m ′ = δ l ′ l δ m ′ E a lm a ∗ m C l , − l ≤ m ≤ l , Thus, E | a lm | 2 = C l , m = 0 , ± 1 , ..., ± l . The series { C 1 , C 2 , ..., C l , ... } is called the angular power spectrum of the isotropic random field T ( θ, ϕ ) . A.Olenko SPDEs on sphere 11 / 32

Random spherical hyperbolic diffusion The papers Anh, V., Broadbridge, P., Olenko, A., Wang, Y. (2018) On approximation for fractional stochastic partial differential equations on the sphere. Stoch. Environ. Res. Risk Assess. 32 , 2585-2603. Broadbridge, P., Kolesnik, A.D., Leonenko, N., Olenko, A. (2019) Random spherical hyperbolic diffusion. Journal of Statistical Physics. 177 , 889-916. Broadbridge, P., Kolesnik, A.D., Leonenko, N., Olenko, A., Omari, D. (2020) Analysis of spherically restricted random hyperbolic diffusion, arXiv:1912.08378, will appear in Entropy. investigated three SPDEs with random initial condition given by CMB. A.Olenko SPDEs on sphere 12 / 32

Model 1: The fractional SPDE on S 2 d X ( t , x ) + ψ ( − ∆ S 2 ) X ( t , x ) = dB H ( t , x ) , t ≥ 0 , x ∈ S 2 , where the fractional diffusion operator ψ ( − ∆ S 2 ) := ( − ∆ S 2 ) α/ 2 ( I − ∆ S 2 ) γ/ 2 is given in terms of Laplace-Beltrami operator ∆ S 2 on S 2 . B H ( t , x ) is a fractional Brownian motion on S 2 with Hurst index H ∈ [1 / 2 , 1) . This equation is solved under the initial condition X (0 , x ) = u ( t 0 , x ), where u ( t 0 , x ), t 0 ≥ 0, is the solution of the fractional stochastic Cauchy problem at time t 0 : ∂ u ( t , x ) + ψ ( − ∆ S 2 ) u ( t , x ) = 0 ∂ t u (0 , x ) = T 0 ( x ) . A.Olenko SPDEs on sphere 13 / 32

Model 2: The hyperbolic diffusion equation ∂ 2 q ( x , t ) 1 + 1 ∂ q ( x , t ) x ∈ R 3 , t ≥ 0 , D > 0 , c > 0 , = ∆ q ( x , t ) , c 2 ∂ t 2 D ∂ t subject to the random initial conditions: � ∂ q ( x , t ) � q ( x , t ) | t =0 = η ( x ) , = 0 , � ∂ t � t =0 where ∆ is the Laplacian in R 3 and η ( x ) = η ( x , ω ) , x ∈ R 3 , ω ∈ Ω is the random field. We investigated T H ( x , t ) , x ∈ S 2 , t > 0 , which is a restriction of the spatial-temporal hyperbolic diffusion field q ( x , t ) to the sphere S 2 . A.Olenko SPDEs on sphere 14 / 32

Model 3: Hyperbolic diffusion equation on the sphere ∂ 2 u ( θ, ϕ, t ) 1 + 1 ∂ u ( θ, ϕ, t ) = k 2 ∆ ( θ,ϕ ) u ( θ, ϕ, t ) , c 2 ∂ t 2 ∂ t D θ ∈ [0 , π ) , ϕ ∈ [0 , 2 π ) , t > 0 , where ∆ ( θ,ϕ ) is the Laplace-Beltrami operator on the sphere ∂ 2 1 ∂ � sin θ ∂ � 1 ∆ ( θ,ϕ ) = + sin 2 θ ∂ϕ 2 sin θ ∂θ ∂θ The random initial conditions are determined by the isotropic random field on the sphere � u ( θ, ϕ, t ) t =0 = T ( θ, ϕ ) , � � ∂ u ( θ, ϕ, t ) � = 0 . � ∂ t � t =0 A.Olenko SPDEs on sphere 15 / 32

Theorem 1 The random solution u ( θ, ϕ, t ) of the initial value problem is � ∞ l − c 2 t � � � u ( θ, ϕ, t ) = exp Y lm ( θ, ϕ ) ξ lm ( t ) , t ≥ 0 , 2 D l =0 m = − l where � 4 π 2 l + 1 a lm Y ∗ ξ lm ( t ) = l 0 ( 0 )[ A l ( t ) + B l ( t )] are stochastic processes with c 2 � � A l ( t ) = cosh ( tK l ) + sinh ( tK l ) 1 � √ � D 2 k 2+ c 2 − Dk 2 DK l l ≤ 2 Dk and c 2 � �� � tK ′ � � tK ′ B l ( t ) = cos + sin 1 � √ � . l l 2 DK ′ D 2 k 2+ c 2 − Dk l > l 2 Dk A.Olenko SPDEs on sphere 16 / 32

The covariance function of the random solution u ( θ, ϕ, t ) is given by − c 2 � � R (cos Θ , t , t ′ ) = Cov ( u ( θ, ϕ, t ) , u ( θ ′ , ϕ ′ , t ′ )) = exp 2 D ( t + t ′ ) ∞ × (4 π ) − 1 � (2 l + 1) C l P l (cos Θ)[ A l ( t ) A l ( t ′ ) + B l ( t ) B l ( t ′ )] , l =0 where Θ = Θ PQ is the angular distance between the points ( θ, ϕ ) and ( θ ′ , ϕ ′ ) and P l ( · ) is the l -th Legendre polynomial, i.e. d l 1 dx l ( x 2 − 1) l . P l ( x ) = 2 l l ! A.Olenko SPDEs on sphere 17 / 32

Convergence study of approximate solutions The approximation of truncation degree L ∈ N to the solution is � L − 1 l − c 2 t � � � u L ( θ, ϕ, t ) = exp Y lm ( θ, ϕ ) ξ lm ( t ) . 2 D l =0 m = − l Theorem 2 For t > 0 the truncation error is bounded by � ∞ � 1 / 2 � � u ( θ, ϕ, t ) − u L ( θ, ϕ, t ) � L 2 (Ω × S 2 ) ≤ C (2 l + 1) C l . l = L √ D 2 k 2 + c 2 − Dk Moreover, for L > it holds 2 Dk � � ∞ � 1 / 2 − c 2 t � � � u ( θ, ϕ, t ) − u L ( θ, ϕ, t ) � L 2 (Ω × S 2 ) ≤ C exp (2 l + 1) C l . 2 D l = L A.Olenko SPDEs on sphere 18 / 32

Corollary 3 Let the angular power spectrum { C l , l = 0 , 1 , 2 , ... } of the random field T ( θ, ϕ ) from the initial condition decay algebraically with order α > 2 , i.e. C l ≤ C · l − α for all l ≥ l 0 . Then, √ D 2 k 2 + c 2 − Dk (i) for L > max( l 0 , ) the truncation error is bounded by 2 Dk − c 2 t � � L − α − 2 2 , � u ( θ, ϕ, t ) − u L ( θ, ϕ, t ) � L 2 (Ω × S 2 ) ≤ C exp 2 D (ii) for any ε > 0 it holds � − c 2 t / D � ≤ C exp � � | u ( θ, ϕ, t ) − u L ( θ, ϕ, t ) | ≥ ε , P L α − 2 ε 2 (iii) for all θ ∈ [0 , π ) , ϕ ∈ [0 , 2 π ) and t > 0 it holds | u ( θ, ϕ, t ) − u L ( θ, ϕ, t ) | ≤ L − β P − a . s ., 0 , α − 3 � � where β ∈ and α > 3 . 2 A.Olenko SPDEs on sphere 19 / 32